We will be working over an algebraically closed field of characteristic 0. We say that a projective variety $X\subset \mathbb{P}^n$ has projectively isomorphic plane sections if there is an open set $U\in(\mathbb{P}^n)^\vee$ such that the hyperplane sections $H\cap X,\ H\in U$ are all projectively isomorphic, i.e. for any two $H_1,H_2\in U$ there is some automorphism $f$ of $\mathbb{P}^n$ such that $f(H_1)=H_2, f(X\cap H_1)=X\cap H_2$.

Consider a smooth cubic hypersurface $X\subset \mathbb{P}^3.$ According to Theorem 2.12 of the paper "Some Remarks on Varieties with Projectively Isomorphic Hyperplane Sections" by Rita Pardini, $X$ has projectively isomorphic hyperplane sections if and only if it's a projection of a normal rational ruled surface in $\mathbb{P}^4.$ I suspect that this is not the case in this situation.

Is there a way to see that $X$ does not have projectively isomorphic hyperplane sections? More generally, what are the ways to identify such projections of normal rational ruled surfaces of degree $d$ from $\mathbb{P}^{d+1}$ to $\mathbb{P}^d$?